#413411

order	512
nilpotency class	3
isoclinism	isoclinic class of #395242
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

762 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2)(3,8)(4,6)(5,7)(9,10,12,14)(11,15,16,13)(17,18,21,25)(19,20,29,23)(22,26,31,27)(24,32,30,28);(1,3,5,4)(2,6,7,8)(9,11,12,16)(10,13,14,15)(17,19,21,29)(18,23,25,20)(22,24,31,30)(26,28,27,32);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(17,18)(19,28)(20,30)(21,25)(22,27)(23,24)(26,31)(29,32);(17,20,22,28,21,23,31,32)(18,24,27,29,25,30,26,19)

Polycyclic presentation:

2^9|a^2=f,b^2=e,c^2=ef,d^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,g^b=gi,d^c=di,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 320607);